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element y of C, we have xO*) = xCvX and there exists a character % such that p(x*x) =
X(x*x). If g is the restriction of % to C, we have \g(y)\ ^p(y) for all y E C. Now use
Section 15.4, Problem 18(e); Section 12.15, Problem 4; and Problem 2(c) above).

(b)    Conversely, if for each .x e A there exists a positive linear form /on A such that
/O) = 1 and/O**)1/2 = p(x\ then A is hermitian. (If a = p(x*x) and u — ae — x*x,
show that /(#2) ^ a2 for all positive linear forms / on A such that f(e)— 1, and
deduce from the hypothesis that p(u2) ^ a2, so that p(oc.e — x*x) <i a ; then use
Section 15.4, Problem 19.)

(c)    Suppose that A is hermitian, and let S(A) be the subspace of the dual A' of the
Banach space A, consisting of the positive linear forms / on A such that f(e) = 1 .
Then S(A) is compact and metrizable for the weak topology (12.15.9). Let (/„) be a
dense sequence in S(A), and let $*-* Ģ/„(•?) be the representation of A on a separable
Hilbert space Hn constructed from /„ by the procedure of (15.6.10). We obtain a
representation U of A on the Hilbert space H which is the Hilbert sum of the Hn
(6.4) by putting U(s) • Z •*ŧ = Z WJ) " *Ŧ • Show that II u(s) II = XA and hence that the

kernel of U is the radical 91 of A, and U is continuous.

(d)    A star algebra A with unit element is hermitian (1 5.4.1 2), and for each x e A we
have \\x\\— p(x) (Section 15.4, Problem 4(a)). If A is separable, deduce that there
exists an isometric isomorphism U of A onto a closed involutive subalgebra of JS?(E)
(where E is a separable Hilbert space), such that U(s*) = (U(s))* for all s e A.

(e)    If E is a separable Hilbert space, the algebra A = C • 1E Đ -^(E) is a hermitian
Banach algebra relative to the norm ||Ģ • 1E + w||2 = |f | H- \\u\\z, in the notation of
(15.4.8). Compare ||w||2 and p(u) for u e ^f2(E).

4.    Let A be a Banach algebra with unit element, and let #i->jc* be a (not necessarily
continuous) involution on A. Show that the following properties are equivalent:

(a) There exists a norm on A, equivalent to the given norm, relative to which A
is a star algebra.

(j3)   There exists a number A> 0 such that ||jc*jt|| S> A||**|| - \\x\\ for all x e A.

(y)   The set of unitary elements of A is bounded.

(3) The involution XH>X* is hermitian, and there exists a number ju,>0 such
that ||a|| ^ /x • p(a) for all hermitian elements a e A.

(To show that (8) implies (a), show that (8) implies that \\x\\ <^ 2fjup(x)t by writing
x = a + ib with a, b hermitian, and using Section 15.4, Problem 18(d) and 18(g).
To show that (]8) implies (8), show first that if a is hermitian, then A||fl|| ^p(a\ by
using (15.2.7); deduce that if x is normal, then A2||xŧ||- \\(x*?\\ <p(x*x)n for all
integers n > 1, and use Section 15.4, Problem 19 to show that A is hermitian. Finally,
to show that (y) implies (S), observe that for each hermitian element a such that
p(a) < I there exists in A a hermitian element 6, commuting with a, such that b2 =
e — a2 (Section 15.4, Problem 17); consequently u — a + ib is unitary, and a =
i(Ŧ H- Ŧ*).)

5.    Let A be a Banach algebra with unit element e, endowed with a hermitian involution

(a)    Let  xeA be  such that p(x)< 1.  Show that x(e-~x*x)1'2 Ŧ (*- xx*)1/2x
(cf. Section 15.2, Problem 11(0).

(b)    Under the same hypotheses, show that the function

= 0? - xx*)-ll\Ģe + x)(e + Ģx*r1(e - x*x)li*ntain a neighborhood of e in H, which would